They won't know but they will make sure you will tell them and they will force you, they'll require you to explain where the money come from and what activities you do on unregistered exchanges otherwise your money will be seized or frozen. They requires sender ID and recipient ID exactly to identify every activity you're doing on the blockchain which defeats the purpose of the blockchain.
If you did bad and illegal things, it makes sense if centralized exchanges take action to terminate accounts based on their data collection and detection on your fraudulent activities on their platforms. They do that for their business safety first, even before government send any specific orders to their centralized exchanges.
However, it's completely different and contains more risks for users if centralized exchanges take advantage of such rules to play against their users, just to terminate user accounts, freeze user funds, then steal it.
https://www.cryptowisser.com/exchange-graveyard/When government regulation comes in, it's guaranteed that they'll try to profile you and your activities on the blockchain because they always want to know about you for tax purpose of course and other motives.
When governments want to do regulations, they can change, and upgrade their laws, acts, regulations with time. When there are such existing regulatory ones, if governments feel like it's time for tightening enforcement, they will do that very quickly and harshly.
Be careful using Binance & Centralized Exchange.